Relativity 107f: General Relativity Basics - Einstein Field Equation Derivation (w/ sign convention)

Einstein Papers:

0:00 Overview of Derivation
6:42 Metric Compatibility + Cosmological Constant term
12:53 Contracted Bianchi Identity
20:54 Solving for Kappa (Einstein Constant)
28:08 Trace-Reversed Form
29:46 Sign Conventions
35:09 Summary

[arshadali]

You're in a position to publish all your Youtube lectures in book form -- color pictures on glossy paper. In 40 years of looking at books on differential geometry, Riemannian geometry, and general relativity, I have never seen anything comparable.

[CallOFDutyMVP]

Amazing work eigenchris. Truly Legendary. I've probably watched 30 hours+ of your videos, between the Tensor Calculus series and Relativity. I was just watching Tensor Calculus 14 today for review. All those hours of notes I have, truly worth it, your videos have changed my outlook on everything. Thank you for everything.

[eigenchris]

ADDITIONAL NOTES AND ERRORS:

Error at 23:18... I accidentally wrote the inner product of vectors instead of the outer product. Whoops.

[teaer]

I finished Tensor Calculus series and General Relativity series 107 videos.
Thank you for your excellent works.
I was a physics student decades ago, and had been working as software engineer after graduation. Last year, I retired, I returned to study physics, in particular General Relativity.

[mohammedkhan]

Hands down, One of the best GR videos on youtube!! Takes 8 years of GR insight and sums it up in 37 mins.

[Handelsbilanzdefizit]

I really appreciate, that you put arrow-signs over your vectors. This makes it much easier to follow.
Many explainations lead to a mess of indices. Most tutors make it unnecessary complex.

[AndreaPancia]

It's amazing what you have been able to summarize and explain in a way many may attempt to understand, far better than what they have struggled to do up to now. Thanks a lot.

[davidwagner]

Thanks Eigenchris, l think i finally get it! I'm 60 and have wanted to know the details for ages.

[michaelzumpano]

I only watched the first 6 minutes tonight, but you gave an incredible intuitive insight into the construction of the field equations. I really love your videos.

[PriyanshuAnand]

Thank you Eigenchris. Your video helped me drastically for my research paper on special and general relativity. Truly grateful to you.

[michaelperrone]

Thanks for uploading all these very instructive videos about how general relativity works: it has been extremely instructive and helpful

[walter--]

Your voice is so monotone and still I keep watching, listening. There is still hope for me...

[boukharroubamediane]

your videos are as beautiful as the identity of Euler "e[^(i*pi)]+1=0" 😊.
They are nicely clear and well explained!!
your efforts in preparing those videos are very grateful. Thanks very much to you and your coworker J. Murray
Thus, I subscribe, like and share. Good lucks.

[William]

This series of video and the tensor calculus one are truly marvellous and are explaining things thoroughly in a nice way with sufficient context to understand things clearly. I am just commenting that in this video it would be better to at least mention the Lovelock’s theorem because that theorem tells us the Einstein tensor is really the unique choice (barring some constant). I myself learned about the Lovelock’s theorem when I found myself unsatisfied with the explanation that since the covariant derivative needs to be zero, we must add the -Rg/2 term, and I was wondering whether there are other tensors that would work as well, and then I found the Lovelock’s theorem, which kinds of rules other possibilities out. I believe I’m not the only one that would have this kind of question and mentioning Lovelock’s theorem would make this already marvellous video even more perfect.

Thank you again for all your great efforts in creating these wonderful video series.

[leonardp]

Excellent work as always, I am hyped for the next videos!! Maybe you can answer therse questions I haven't found answers to online:
Is there any intuitive geometric interpretation of the Einstein tensor, like how one can interpret the components of the Ricci tensor as geodesic deviation?
Also, what kind of surfaces give a nonzero Einstein tensor? I have tried the sphere, saddle, and a few other random ones, but everything I tried produces an all-zero Einstein tensor.

[sebastiandierks]

Maybe you shouldn't call it a "derivation", more a justification/plausibility check on theoretical grounds. In the end, experiments had to show that it's the "correct" theory of gravity (within its range of applicability). You cant really derive the most fundamental equations of physics like the standard model Lagrangian etc. Anyway, that's just semantics, amazing video with a huge information density.

[narfwhals]

This was a mouthful. Thank you very much!
The small perturbation of the metric linearizes the field equations, right? I assume you're going to go into that a bit in the video on gravitational waves.
I'm really interested in the differences of linear and non-linear physics recently.
I'm not really sure where I'm going with this so lets just say I'm commenting for the algorithm.

[asmaiqbal]

your videos are really informative and interesting. please make some videos on basic understanding of quantum mechanics too

[MessedUpSystem]

Wasn't there a way to derive the Einstein Field Equations through extremal action? But I guess it would be significantly harder anyway

[ToddDesiato]

That was really well done. Thank you. Did I understand correctly? That, the only justification for equating kappa*T^uv with G^uv, was because their covariant derivatives were both zero. There had to be more to it than that. Einstein could've simply put the cosmological constant term alone on the LHS and it would've satisfied that condition. I prefer the interpretation where, T^uv tells us all the forces acting on matter at that location, and G^uv is simply what we measure using deformed rulers and clocks.